Multiplicity of Positive Solutions for a Quasilinear Schrödinger Equation with an Almost Critical Nonlinearity

Abstract

Abstract In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: { - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = | u | p - 2 ⁢ u in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u|% ^{p-2}u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth and bounded domain in ℝ N , N ≥ 3 {\mathbb{R}^{N},N\geq 3} . More specifically we prove that, for p near the critical exponent 22 * = 4 ⁢ N / ( N - 2 ) {22^{*}=4N/(N-2)} , the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.